MTH-102复习2

随机变量上半部分，包括基础内容和离散随机变量

Lec5

1. Random variables

   1. Definition

      Given a random experiment with a sample space S, a function X that assigns one and only one real number X(s) = x to each element s in S is called a random variable. The space of X is the set of real numbers
      where means that the element s belongs to the set S
      随机变量X将S样本空间中的抽象的概念用数字表现出来。

   2. Probability mass function (pmf)

      Let X be a discrete random variable and be the values that X can take on. The pmf p(x) is a function that satisfies the following properties:
      1.
      2.
      3. , for any event A.
      pmf指概率质量函数。在概率论中，概率质量函数是离散随机变量在各特定取值上的概率
      例如，投掷一枚硬币，令：那么，其PMF为

   3. Cumulative distribution function (cdf)

      • We call the function defined bythe cumulative distribution function and abbreviate it as cdf
      • Let be the pmf of the random variable , and be the values that can take on. Then for any
      • For any ,
      • is a nondecreasing function, i.e. for .
      • 同样是上面的硬币问题，其cdf为

      • Properties of cdf

        1. The cdf is right continuous, i.e. for any where is the right limit of at
        2. For any ,where is the left limit of at
        3. For ,

2. Mean and variance

   1. Definition of mean

      If is a discrete random variable having a probability mass function , then the mean, or the expectation, of , denoted by , is defined byiIf the values that can take on are , thenIn words, the mean of is a weighted average of the possible values that can take on, each value being weighted by the probability that assumes it.

   2. Expectation of a function of a random variable

      If X is a discrete random variable that takes on the values , with the pmf p(x), then for any real-valued function Remark is a actually a new random variable. However, to compute the mean of , i.e. , it is not necessary to find the distribution of once the distribution of is given.

   3. Definition of variance

      The variance of a random variable , denoted by , is defined as the mean of the function of with , i.e. In the practice, it is more convenient to compute the variance vir the following equivalent formula
      • The variance is always nonnegative
      • The square root of , i.e. , is called the standard deviation of

   4. Properties of mean and variance

      Let be a random variable, , and be constants. Consider as a linear function of . Then
      The formula can be proved using the above properties
      • Let be random variables. The mean of a sum of random variables equals the sum of the mean of each random variable, i.e.Moreover, let be constant. Then

Lec6

离散随机变量(Discrete random)指的是一些值取得不连续的随机变量，上述所有的定义和概念都建立在随机变量是离散随机变量的情况下。
下面会介绍一些简单的离散随机变量

1. Bernoulli distribution

   • A Bernoulli experiment is a random experiment with two outcomes, modeled with the sample space
   • Let be a random variable associated with a Bernoulli experiment with for some . The pmf of can be written as
   • We say that has a Bernoulli distribution, and

2. Binomial distribution

   • A Bernoulli experiment is performed times independently, and let the random variable be the number of times when the outcome is 1 in the trials
   • The sample space of is
   • The pmf of is
   • X is said to have a binomial distribution, which is denoted by the symbol . The constants and are called the parameters of the binomial distribution
   • A Bernoulli distribution is just a binomial distribution with parameters
   • If a random variable has a binomial distribution with parameters , then The mean and variance can be computed in two ways
     1. Direct computation with series:
     2. For , let be the outcome of the i-th trial. Then has a Bernoulli distribution () and Therefore,

3. Geometric distribution

   • Motivation: we are interested in the number of independent trials needed in order to get the first success. The probability of success in each trial is constantly
   • The sample space
   • A random variable is said to have a geometric distribution if the pmf of is defined bywhere
   • The cdf of a geometric random variable is
   • The mean of is
   • The variance of is

4. Poisson distribution

   • The pmfwhere
   • The mean of is
   • The variance of is
   • 通常为数据的均值
   • 泊松分布是由二项分布推导得来。但二项分布的p很小时，两者比较接近